https://github.com/tancik/fourier-feature-networks

Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains
https://github.com/tancik/fourier-feature-networks

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Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains

• Host: GitHub
• URL: https://github.com/tancik/fourier-feature-networks
• Owner: tancik
• License: mit
• Created: 2020-06-16T23:54:50.000Z (over 4 years ago)
• Default Branch: master
• Last Pushed: 2023-01-17T21:33:18.000Z (almost 2 years ago)
• Last Synced: 2024-12-15T08:03:18.817Z (10 days ago)
• Language: Jupyter Notebook
• Homepage: https://people.eecs.berkeley.edu/~bmild/fourfeat/
• Size: 12.3 MB
• Stars: 1,269
• Watchers: 25
• Forks: 134
• Open Issues: 12
• Metadata Files:
  • Readme: README.md
  • License: LICENSE

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README

        
# Fourier Features Let Networks Learn  High Frequency Functions in Low Dimensional Domains

### [Project Page](https://bmild.github.io/fourfeat/) | [Paper](https://arxiv.org/abs/2006.10739)

[![Open Demo in Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/tancik/fourier-feature-networks/blob/master/Demo.ipynb)


[Matthew Tancik](http://tancik.com/)\*1,

[Pratul P. Srinivasan](https://people.eecs.berkeley.edu/~pratul/)\*1,2,

[Ben Mildenhall](https://people.eecs.berkeley.edu/~bmild/)\*1,

[Sara Fridovich-Keil](https://people.eecs.berkeley.edu/~sfk/)1,

[Nithin Raghavan](https://www.linkedin.com/in/nithinraghavan/)1,

[Utkarsh Singhal](https://scholar.google.com/citations?user=lvA86MYAAAAJ&hl=en)1,

[Ravi Ramamoorthi](http://cseweb.ucsd.edu/~ravir/)3,

[Jonathan T. Barron](http://jonbarron.info/)2,

[Ren Ng](https://www2.eecs.berkeley.edu/Faculty/Homepages/yirenng.html)1


1UC Berkeley, 2Google Research, 3UC San Diego 


*denotes equal contribution

## Abstract

![Teaser Image](https://user-images.githubusercontent.com/3310961/84946597-cdf59800-b09d-11ea-8f0a-e8aaeee77829.png)

We show that passing input points through a simple Fourier feature mapping enables a multilayer perceptron (MLP) to learn high-frequency functions in low-dimensional problem domains. These results shed light on recent advances in computer vision and graphics that achieve state-of-the-art results by using MLPs to represent complex 3D objects and scenes. Using tools from the neural tangent kernel (NTK) literature, we show that a standard MLP fails to learn high frequencies both in theory and in practice. To overcome this spectral bias, we use a Fourier feature mapping to transform the effective NTK into a stationary kernel with a tunable bandwidth. We suggest an approach for selecting problem-specific Fourier features that greatly improves the performance of MLPs for low-dimensional regression tasks relevant to the computer vision and graphics communities.

## Code

We provide a [demo IPython notebook](https://colab.research.google.com/github/tancik/fourier-feature-networks/blob/master/Demo.ipynb) as a simple reference for the core idea. The scripts used to generate the paper plots and tables are located in the [Experiments](https://github.com/tancik/fourier-feature-networks/tree/master/Experiments) directory.